Properties

Label 21.3.426...007.1
Degree $21$
Signature $[3, 9]$
Discriminant $-4.267\times 10^{27}$
Root discriminant $20.69$
Ramified primes $7, 184607$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 21T56

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 7*x^19 + 6*x^18 + 16*x^17 - 27*x^16 + 3*x^15 + 119*x^14 + 23*x^13 - 181*x^12 - 153*x^11 + 154*x^10 + 417*x^9 - 133*x^8 - 395*x^7 + 153*x^6 + 61*x^5 - 33*x^4 + 2*x^3 + 7*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^21 - x^20 - 7*x^19 + 6*x^18 + 16*x^17 - 27*x^16 + 3*x^15 + 119*x^14 + 23*x^13 - 181*x^12 - 153*x^11 + 154*x^10 + 417*x^9 - 133*x^8 - 395*x^7 + 153*x^6 + 61*x^5 - 33*x^4 + 2*x^3 + 7*x^2 - 2*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -33, 61, 153, -395, -133, 417, 154, -153, -181, 23, 119, 3, -27, 16, 6, -7, -1, 1]);
 

\( x^{21} - x^{20} - 7 x^{19} + 6 x^{18} + 16 x^{17} - 27 x^{16} + 3 x^{15} + 119 x^{14} + 23 x^{13} - 181 x^{12} - 153 x^{11} + 154 x^{10} + 417 x^{9} - 133 x^{8} - 395 x^{7} + 153 x^{6} + 61 x^{5} - 33 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-4266945093472415611428661007\)\(\medspace = -\,7^{14}\cdot 184607^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.69$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 184607$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{450670194838983861374947} a^{20} + \frac{56223861795849143322841}{450670194838983861374947} a^{19} + \frac{64249336049343100829494}{450670194838983861374947} a^{18} + \frac{204399107601236243042313}{450670194838983861374947} a^{17} + \frac{3642335875687647116509}{10991955971682533204267} a^{16} - \frac{21419957338147771651350}{450670194838983861374947} a^{15} + \frac{200992861462069205358110}{450670194838983861374947} a^{14} - \frac{69822989021112760143602}{450670194838983861374947} a^{13} + \frac{118966102136775017940697}{450670194838983861374947} a^{12} - \frac{123218055012213369623944}{450670194838983861374947} a^{11} - \frac{123540447366436114592452}{450670194838983861374947} a^{10} + \frac{11146526721641761557556}{450670194838983861374947} a^{9} - \frac{37421128641923606927800}{450670194838983861374947} a^{8} - \frac{71535673557524879023008}{450670194838983861374947} a^{7} - \frac{105955011820037945569376}{450670194838983861374947} a^{6} - \frac{175183607652817456057904}{450670194838983861374947} a^{5} - \frac{69323942590536381769902}{450670194838983861374947} a^{4} - \frac{103398456252914015991171}{450670194838983861374947} a^{3} - \frac{201083225688327238344797}{450670194838983861374947} a^{2} - \frac{156767888372066062009932}{450670194838983861374947} a - \frac{38107391571513227497902}{450670194838983861374947}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 201933.335135 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 201933.335135 \cdot 1}{2\sqrt{4266945093472415611428661007}}\approx 0.188724506101$ (assuming GRH)

Galois group

21T56:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 15120
The 45 conjugacy class representatives for t21n56
Character table for t21n56 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.184607.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 42 sibling: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $21$ $21$ $15{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R $15{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $15{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $21$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
184607Data not computed